Mathematical Optimisation - Fundamentals and Applications
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The document provides an overview of mathematical optimisation, detailing its applications in various fields such as manufacturing, finance, and engineering, alongside its historical development from linear programming to more complex forms like semidefinite programming. It also covers fundamental concepts, problem types, and practical examples of optimisation in areas including data fitting and portfolio optimisation. The text highlights the importance of convex optimisation, robust methods, and advancements in computational techniques for solving complex optimisation problems.
![WHAT IS
MATHEMATICAL
OPTIMISATION
FORMALLY, MATHEMATICAL
OPTIMISATION IS THE PROCESS OF
FORMULATION AND SOLUTION OF
A CONSTRAINED OPTIMISATION PROBLEM
OF THE GENERAL MATHEMATICAL FORM:
MINIMIZE F(X), X = [X1, X2, X3, …. ] T E R N
SUBJECT TO THE CONSTRAINTS
G(X) < 0 , J = 1, 2, … , M
H(X) = 0 , J = 1, 2, …, R
WHERE F(X), G(X), AND H(X) ARE SCALAR
FUNCTIONS OF THE REAL COLUMN VECTOR
X.](https://image.slidesharecdn.com/mathematicaloptimization-industryperspective-171226071728/85/Mathematical-Optimisation-Fundamentals-and-Applications-10-320.jpg)
![MATHEMATICAL OPTIMISATION
OBJECTIVE AND CONSTRAINT FUNCTIONS
• The values of the functions f(x), g(x),
and h(x) at any point x = [x1, x2,
…xn] may in practice be obtained in
different ways
• From analytically known formulae
• as the outcome of some
computational process, g(x) = a(x)
- amax
• From measurement taken from a
physical process, e.g. h1(x) = T(x)
- T0](https://image.slidesharecdn.com/mathematicaloptimization-industryperspective-171226071728/85/Mathematical-Optimisation-Fundamentals-and-Applications-11-320.jpg)
Mathematical Optimisation - Fundamentals and Applications
- 1. MATHEMATICAL OPTIMISATION AN INTRODUCTION TO THE INDUSTRIAL APPICATIONS OF
- 2. KNOWN AS NON-LINEAR PROGRAMMING OR NUMERICAL OPTIMISATION OVERVIEW
- 3. INTRODUCTION MATHEMATICAL OPTIMISATION IN THE REAL WORLD • Manufacturing • Production • Inventory Control • Transportation • Scheduling • Networks • Finance • Control Engineering • Marketing • Policy Modelling
- 4. A NARRATIVE MATHEMATICAL OPTIMISATION IN THE REAL WORLD • Optimising time in the production cycle of an industry, optimising tax in a tax return, optimising length in a tour are mathematical optimisation problems we encounter in our daily life. • A solution that satisfying all the constraints is called a feasible solution • The set of all solutions, satisfying all the constraints is called the feasible region
- 5. A SNAPSHOT HISTORY • 1940s • Linear Programming • 1950s • Quadratic Programming • 1960s • Geometric Programming • 1990s • Semidefinite Programming • Quadratically Constrained Quadratic Programming • Robust Optimizaton • Sum of Squares Programming
- 6. A QUICK SURVEY NEW APPLICATIONS SINCE 1990 • Linear matrix inequality techniques in control • Circuit design via geometric programming • Support vector machine learning via quadratic programming • Semidefinite programming relaxations in combinatorial optimisation • L1- norm optimisation for sparse signal reconstruction • Applications in quantum information theory, computer vision, image processing, finance
- 7. MATHEMATICAL OPTIMISATION MANY OTHER KINDS … • Linear Network Optimization • Specialization of LP to problems with graph structure ( shortest path, transportation, trans-shipment, etc.) • Integer Programming • Decision variables are allowed only integer values • Combinatorial optimisation • Decision variables have nice combinatorial structures ( e.g. trees, permutations, matchings )
- 8. DEEPER INSIGHTS MATHEMATICAL OPTIMISATION : SPECIAL CASES • Optimal solution need not be unique • One of the special case is when variables have symmetry. • In this case, some kind of permutation can be applied to get multiple optimal solution
- 9. MATHEMATICAL OPTIMISATION ESSENTIAL NOTIONS • Design Variables • Objective Functions • Inequality constraint functions • Equality constraint functions • Optimum Vector
- 10. WHAT IS MATHEMATICAL OPTIMISATION FORMALLY, MATHEMATICAL OPTIMISATION IS THE PROCESS OF FORMULATION AND SOLUTION OF A CONSTRAINED OPTIMISATION PROBLEM OF THE GENERAL MATHEMATICAL FORM: MINIMIZE F(X), X = [X1, X2, X3, …. ] T E R N SUBJECT TO THE CONSTRAINTS G(X) < 0 , J = 1, 2, … , M H(X) = 0 , J = 1, 2, …, R WHERE F(X), G(X), AND H(X) ARE SCALAR FUNCTIONS OF THE REAL COLUMN VECTOR X.
- 11. MATHEMATICAL OPTIMISATION OBJECTIVE AND CONSTRAINT FUNCTIONS • The values of the functions f(x), g(x), and h(x) at any point x = [x1, x2, …xn] may in practice be obtained in different ways • From analytically known formulae • as the outcome of some computational process, g(x) = a(x) - amax • From measurement taken from a physical process, e.g. h1(x) = T(x) - T0
- 12. INTRODUCTION MATHEMATICAL OPTIMISATION : PROBLEM TYPES • Limited or Unlimited • One variable or Many variables • Discrete Variable or Continuous Variables • Static Problems or Dynamic Problems • Deterministic or Stochastic Problems • Linear Equations or Nonlinear Equations
- 13. MATHEMATICAL OPTIMISATION SENSITIVITY AND ROBUSTNESS • Every model we write is only a coarse description of reality • Conclusions about the model may more may not correspond to actual behaviour • Validity of models often informal, implicit • These considerations sometimes incorporated through sensitivity analysis
- 14. MATHEMATICAL OPTIMISATION SENSITIVITY AND ROBUSTNESS • More recently, better techniques to explicit account of difference between real world and model ( robust control, robust optimisation ) • Constraints and / or objective known only approximately • Implemented solution different from the computed one
- 15. MODEL CLASSIFICATIONS MATHEMATICAL OPTIMISATION • Unconstrained Optimisation • Linear Optimisation • Linear Constrained Optimisation : • If the constraint functions are linear / affine • Conic Linear Optimisation : • If both the objective and the constraint functions are linear / affine but variables in a convex cone • Quadratically constrained Quadratic Optimisation : • If both the objective and constraint functions are quadratic
- 16. MATHEMATICAL OPTIMISATION GLOBAL OPTIMISATION • Consider unconstrained optimisation • Typically extremely difficult, many local minima • Many questions can be posed in these terms ( protein folding ) • Very flexible formulation • But hard to do anything substantial with it • Complexity theoretic obstacles
- 17. BASIC OVERVIEW CONVEX OPTIMISATION • Convex optimisation is a generation of linear programming where the constraints and object functions are convex. • Many subclasses of convex optimisation like semidefinite programming and least square problem are also widely used and have important applications in various fields. • This case guarantees that we are able to find global minimum.
- 18. MATHEMATICAL OPTIMISATION CONVEX OPTIMISATION • Objective function is convex • Feasible set is convex • Many advantages • Modelling Flexibility • Tractability and Scalability • Sensitivity Analysis relatively simple • Can naturally incorporate robust considerations
- 19. FUNDAMENTAL CONCEPTS LINEAR PROGRAMMING • Linear programming is one of the well studied classes of optimisation problem. • A linear program is one which has linear objectives and constraint functions. • Examples: • Max flow: Given a graph, start and end node, capacities on every edge, find out the maximum flow possible through edges. • Simplex, Ellipsoid, Interior Point Method are some of the well known algorithms for solving linear programming problems • Simplex method was one of the first methods to solve these programs. But all the initial versions took exponential time • The first polynomial time algorithm was ellipsoid algorithm. Few years later interior point method was developed and shown to be in polynomial time.
- 20. MATHEMATICAL OPTIMISATION OTHER CONVEX OPTIMISATION METHODS • Least Squares Problems • Used for data fitting • Semidefinite programming • Quadratic Programming
- 21. MATHEMATICAL OPTIMISATION SEMIDEFINITE PROGRAMMING • A broad generalisation of LP to symmetric matrices • The intersection of an affine subspace L and the cone of positive semidefinite matrices • Originated in control theory and combinatorial optimisation • Convex finite dimensional optimisation • Essentially solvable in polynomial time
- 22. COMPLEXITY MATTERS NONLINEAR OPTIMISATION • The general optimisation problem is intractable • Even simple looking optimisation problems can be very hard • Examples • Quadratic optimisation problem with many constraints • Minimising a multivariate polynomial
- 23. EMBEDDED OPTIMISATION PRACTICAL EXAMPLES OF USING OPTIMISATION • In the last decades, the size of computers and their components decreased and therefore it became beneficial to optimise the device sizing in electric circuits • For this problem the objective function is a power consumption • The variables are widths and lengths of the device and the constraints are manufacturing limits, timing requirements, and maximum area.
- 24. PORTFOLIO OPTIMISATION PRACTICAL EXAMPLES OF USING MATHEMATICAL OPTIMISATION • When investing in assets there is a big risk of loosing invested money and so it would be a competitive advantage to posses a control system, that would find the most risk free way for your investment • For this purpose, the objective function could be overall risk or return variance, the variables amounts invested in different assets, and the budget, maximum and minimum investment per asset and minimum return would be our constraints.
- 25. DATA FITTING PRACTICAL EXAMPLES OF USING MATHEMATICAL OPTIMISATION • In this case, we are looking for the best fitting model for our observed data. • The objective function can represent the measure of misfit or prediction error. • The variables model parameters and constraints can feature prior information and parameter limits.
- 26. MODEL PREDICTIVE CONTROL PRACTICAL EXAMPLES OF USING MATHEMATICAL OPTIMISATION • This is a concept of advanced control systems such as heating control of intelligent buildings or control of chemical processes • This method works with complex dynamic behaviour of the system and is aiming mainly to minimise the performance criterion in the future that would possibly be subject to constraints • In MPC there are dependent and independent variables
- 27. QUANTUM COMPUTING AND MATHEMATICAL OPTIMISATION MATHEMATICAL OPTIMISATION
- 28. LOREM IPSUM QUANTUM INFORMATION AND MATHEMATICAL OPTIMISATION • Convexity naturally arises in many places in Quantum Information Theory as the possible preparations, processes and measurements for Quantum systems are convex sets • Quantum Error Correction • Quantum Entanglement Estimation • Quantum Tomography
- 29. LOREM IPSUM CONVEX OPTIMISATION FOR QUANTUM ENTANGLEMENT • Using off the shelf semidefinite programming solvers • Measurement of relative entropy of entanglement • Measurement of Rains bond, known upper bound on the distillable entanglement • Identifying the lower bound on the quantum conditional mutual information interns of the relative entropy • At least one subsystem of a multipart state is a quit
- 30. MATHEMATICAL OPTIMISATION IN DIGITAL TECHNOLOGY INFINITYLABS EXPERIMENTS